INTEGRAL MEANS OF THE DERIVATIVES OF BLASCHKE PRODUCTS

INTEGRAL MEANS OF THE DERIVATIVES OF BLASCHKE PRODUCTS EMMANUEL FRICAIN, JAVAD MASHREGHI Abstract. We study the rate of growth of some integral means of the derivatives of a Blaschke product and we generalize several classical results. Moreover, we obtain the rate of growth of integral means of the derivative of functions in the model subspace KB generated by the Blaschke product B. 1. Introduction Let (zn)n≥1 be a sequence in the unit disc satisfying the Blaschke condition (1.1) ∞ ∑ n=1 (1? |zn|) < ∞. Then, the product B(z) = ∞ ∏ n=1 |zn| zn zn ? z 1? z¯n z is a bounded analytic function on the unit disc D with zeros only at the points zn, n ≥ 1, [5, page 20]. Since the product converges uniformly on compact subsets of D, the logarithmic derivative of B is given by B?(z) B(z) = ∞ ∑ n=1 1? |zn|2 (1? z¯n z)(z ? zn) , (z ? D). 2000 Mathematics Subject Classification. Primary: 30D50, Secondary: 32A70. Key words and phrases. Blaschke products, model space. This work was supported by NSERC (Canada) and FQRNT (Quebec).

• blaschke sequence


• zn


• positive continuous


• condition ∑∞


• points zn


• classical results


• function satisfying


• bergman space